The present paper is devoted to three-dimensional $C_{12}$-manifolds (defined by D. Chinea and C. Gonzalez), which are never normal. We study their fundamental properties and give concrete examples. As an application, we study such structures on three-dimensional Lie groups.

Let $R$ be a commutative local principal ideal ring which is not integral, $f$ a polynomial in $R[x]$ such that $f(0) \neq 0$ and $N(f)$ its Newton polygon. If $N(f)$ contains $r$ sides of different slopes, we show that $f$ has at least $r$ different pure factors in $R[x]$. This generalizes the Newton polygon method over a ring which is not integral.

Given an integer $n\ge 2$, let $P(n)$ stand for its largest prime factor. We examine the behaviour of $\sum\limits_{n\le x \atop n\in A} P(n)$ in the case of two sets $A$, namely the set of $r$-free numbers and the set of $h$-full numbers.

We define new classes of statistical manifolds with a complex structure. Motivation for our work is the classification of almost Hermitian manifolds with respect to the covariant derivative of the almost complex structure, obtained by Gray and Hervella in 1980. Instead of the Levi-Civita connection, we use a statistical one and obtain eight classes of Kähler manifolds with the statistical connection. Besides, we give some properties of tensors constructed from covariant derivative of the almost complex structure with respect to the statistical connection. From the obtained properties, further investigation of statistical manifolds is possible.

We determine generators of the coordinate ring of $\mathrm{SL}_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $\mathrm{SL}_2$-character variety. For free groups $F_k$, we find transcendental generators. Finally, for the case of the $2$-torus, we get an explicit equation of the $\mathrm{SL}_2$-character variety and use the description to compute their $E$-polynomials.

Let $\{M_n\}_{n=1}^{\infty}$ be a sequence of expanding positive integral matrices with $M_n= \begin{pmatrix} p_n & 0\\0 & q_n \end{pmatrix}$ for each $n\ge 1$, and let $D=\left\{\begin{pmatrix} 0\ 0 \end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix}, \begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$ be a finite digit set in $\mathbb{Z}^2$. The associated Borel probability measure obtained by an infinite convolution of atomic measures \[ \mu_{\{M_n\},D}=\delta_{M_1^{-1}D}*\delta_{(M_2M_1)^{-1}D}*\cdots*\delta_{(M_n\cdots M_2M_1)^{-1}D}*\cdots \] is called a Moran–Sierpinski-type measure. We prove that, under certain conditions, $\mu_{\{M_n\}, D}$ is a spectral measure if and only if $3\mid p_n$ and $3\mid q_n$ for each $n\geq2$.

This work presents closed formulas for the determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.

We establish, based on variational methods, existence theorems for a $p$-Kirchhoff-type Neumann problem under the Landesman–Lazer type condition and under the local coercive condition. In addition, multiple solutions for a $p$-Kirchhoff-type Neumann problem are established using a known three-critical-point theorem proposed by H. Brezis and L. Nirenberg.

We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of van den Bergh. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds by Chen et al.

Let $A$ be a finite-dimensional $k$-algebra over an algebraically closed field $k$. In this note, we study the Gorenstein homological properties of a split-by-nilpotent extension algebra. Let $R$ be a split-by-nilpotent extension of $A$. We provide sufficient conditions to ensure when a Gorenstein-projective module over $A$ induces a similar structure over $R$. We also study when a Gorenstein-projective $R$-module induces a Gorenstein-projective $A$-module. Moreover, we study the relationship between the Gorensteinness of $A$ and $R$.

Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L}} I$) be the number of distance Laplacian eigenvalues of a graph $G$ which lie in $I$. For a prescribed interval $I$, we give the bounds for $m_{D^{L}}I$ in terms of the independence number $\alpha(G)$, the chromatic number $\chi$, the number of pendant vertices $p$, the number of components in the complement graph $C_{\overline{G}}$ and the diameter $d$ of $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, $m_{D^{L}(G)}[n,n+\alpha(G))\leq n-\alpha(G)$, $m_{D^{L}(G) }[n,n+p)\leq n-p$ and discuss the cases where the bounds are best possible. In addition, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. We also propose some problems of interest.

We consider a sequence of non-negative $\beta$-mixing random variables $\{X,X_n : n\geq1\}$ from the classical St. Petersburg game. The accumulated gains $S_n=X_1+X_2+\cdots+X_n$ in the St. Petersburg game are studied, and the large deviations and the weak law of large numbers of $S_n$ are obtained.

We describe the ideal of relations for the trivial extension $T(\Lambda)$ of a finite-dimensional monomial algebra $\Lambda$. When $\Lambda$ is, moreover, a gentle algebra, we solve the converse problem: given an algebra $B$, determine whether $B$ is the trivial extension of a gentle algebra. We characterize such algebras $B$ through properties of the cycles of their quiver, and show how to obtain all gentle algebras $\Lambda$ such that $T(\Lambda) \cong B$. We prove that indecomposable trivial extensions of gentle algebras coincide with Brauer graph algebras with multiplicity one in all vertices in the associated Brauer graph, result proven by S. Schroll.

We study the extensions of the classical Eneström–Kakeya theorem and its various generalizations regarding the distribution of zeros of polynomials from the complex to the quaternionic setting. We aim to build upon the previous work by various authors and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients, namely quaternionic coefficients whose real and imaginary components or moduli of the coefficients satisfy suitable inequalities. The obtained results for this subclass of polynomials and slice regular functions produce generalizations of a number of results known in the literature on this subject.

Let $R$ be a commutative ring and $Z(R)$ its zero-divisors set. The weakly zero-divisor graph of $R$, denoted by $W\Gamma(R)$, is an undirected graph with the nonzero zero-divisors $Z(R)^*$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exist $a \in \mathrm{Ann}(x)$ and $b \in \mathrm{Ann}(y)$ such that $ab = 0$. In this paper, we characterize finite rings $R$ for which the weakly zero-divisor graph $W\Gamma(R)$ belongs to some well-known families of graphs. Further, we classify the finite rings $R$ for which $W\Gamma(R)$ is planar, toroidal or double toroidal. Finally, we classify the finite rings $R$ for which the graph $W\Gamma(R)$ has crosscap at most two.

Let $M$ and $b$ be integers greater than 1, and let $\mathbf{p}$ be a positive probability vector for the alphabet $\mathcal{A}_{b}=\{0,\ldots,b-1\}$. Let us consider a random sequence $w_0,w_1,\ldots,w_j$ over $\mathcal{A}_{b}$, where the $w_i$'s are independent and identically distributed according to $\mathbf{p}$. Such a sequence represents, in base $b$, the number $n=\sum_{i=0}^j w_i b^{j-i}$. In this paper, we explore the asymptotic distribution of $n\mbox{ mod }M$, the remainder of $n$ divided by $M$. In particular, by using the theory of Markov chains, we show that if $M$ and $b$ are coprime, then $n\mbox{ mod } M$ exhibits an asymptotic discrete uniform distribution, independent of $\mathbf{p}$; on the other hand, when $M$ and $b$ are not coprime, $n\mbox{ mod }M$ does not necessarily have a uniform distribution, and we obtain an explicit expression for this limiting distribution.

Let $R$ be a commutative ring with non-zero identity. The small intersection graph of $R$, denoted by $G(R)$, is a graph with the vertex set $V(G(R))$, where $V(G(R))$ is the set of all proper non-small ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap J$ is not small in $R$. In this paper, we introduce a certain subgraph $PG(R)$ of $G(R)$, called the principal small intersection graph of $R$. It is the subgraph of $G(R)$ induced by the set of all proper principal non-small ideals of $R$. We study the diameter, the girth, the clique number, the independence number and the domination number of $PG(R)$. Moreover, we present some results on the complement of the principal small intersection graph.

The positive approximate Karush–Kuhn–Tucker sequential condition and the strict constraint qualification associated with this condition for general scalar problems with equality and inequality constraints have recently been introduced. In this paper, we extend them to optimization problems with additional abstract set constraints. We also present an extension of the approximate Karush–Kuhn–Tucker sequential condition and its related strict constraint qualification. Furthermore, we explore the relations between the new constraint qualification and other constraint qualifications known in the literature as Abadie, quasi-normality and the approximate Karush–Kuhn–Tucker regularity constraint qualification. Finally, we introduce an augmented Lagrangian method for solving the optimization problem with abstract set constraints and we show that it is possible to obtain global convergence under the new condition.

We give the basic properties of discrete convolution in the space of complexity functions and its dual space. Two inequalities are identified, and defined in the general context of an arbitrary binary operation in any weighted quasi-metric space. In that setting, some quasi-metric and convergence consequences of those inequalities are proven. Using convolution, we show a method for building improver functionals in the complexity space. We also consider convolution in three topologies within the dual space, obtaining two topological monoids.

Let $(X,J,g,\omega)$ be a compact $2n$-dimensional almost-Kähler manifold. We prove primitive decompositions of $\partial$-, $\bar\partial$-harmonic forms on $X$ in bidegree $(1,1)$ and $(n-1,n-1)$ (such bidegrees appear to be optimal). We provide examples showing that in bidegree $(1,1)$ the $\partial$- and $\bar\partial$-decompositions differ.

Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$-module, and $X$ an arbitrary $R$-module. In this paper, we first prove that if $\dim_R(M)\leq{n+2}$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. As a consequence, it follows that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M))$ is a finite set for all $i$ when $R$ is a semi-local ring and $\dim_R(M)\leq{3}$. Then, we show that if $\dim(R/\mathfrak{a})\leq{n+1}$, then $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i$ whenever $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{\dim_R(X)-n}$. Finally, in the case that $\dim(R/\mathfrak{a})\leq{2}$, $X$ is $\mathfrak{a}$-torsion, and $n>0$ or $\operatorname{Supp}_R(X)\cap\operatorname{Var}(\mathfrak{a})\cap\operatorname{Max}(R)$ is finite, we prove that $X$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{2-n}$. We conclude with some ordinary $\mathfrak{a}$-cofiniteness results for local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(X)$.

The object of the present paper is to study $C_{12}$-structures on a manifold with Lorentzian metric. We focus here on Lorentzian $C_{12}$-structures, emphasizing their relationship and analogies with respect to the Riemannian case. Several interesting results are obtained. Next, we study Ricci solitons in Lorentzian $C_{12}$-manifolds.

We generalize the construction of face algebras by Hayashi and obtain a left bialgebroid $\mathfrak{A}(w)$. There are some relations between the left bialgebroid $\mathfrak{A}(w)$ and the generalized Shibukawa–Takeuchi left bialgebroid $A_{\sigma}$.